On the number of limit cycles coming from a uniform isochronous center with continuous and discontinuous quartic perturbations

• Autorzy: Rezaiki Nabil , Boulfoul Amel

Identyfikatory

DOI

10.1515/jaa-2023-0018

• Języki publikacji: EN

Abstrakty

EN

In this paper,westudy the number of limit cycles bifurcated from the periodic orbits of a cubic uniform isochronous center with continuous and discontinuous quartic polynomial perturbations. Using the averaging theory of first order for continuous and discontinuous differential systems and comparing the obtained results, we show that the discontinuous systems have at least 6 more limit cycles than the continuous ones. This study needs some computations that have been verified using Maple.

Słowa kluczowe

EN

averaging theory; continuous differential system; discontinuous differential system; isochronous center; limit cycle

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2024
• Tom: Vol. 30, nr 1
• Strony: 35--50
• Opis fizyczny: Bibliogr. 23 poz., wykr.

Twórcy

autor

Rezaiki Nabil

• LMA Laboratory, Department of Mathematics, University of Badji Mokhtar, P. O. Box 12, Annaba, 23000 Algeria

autor

Boulfoul Amel

• Department of Mathematics, 20 aout 1955 University, BP 26, El Hadaiek 21000, Skikda, Algeria

Bibliografia

• [1] I. S. Berezin and N. P. Shidkov, Computing Methods. Vols. I, II, Pergamon Press, Oxford, 1964.
• [2] A. Buică and J. Llibre, Averaging methods for finding periodic orbits via Brouwer degree, Bull. Sci. Math. 128 (2004), no. 1, 7-22.
• [3] A. G. Choudhury and P. Guha, On commuting vector fields and Darboux functions for planar differential equations, Lobachevskii J. Math. 34 (2013), no. 3, 212-226.
• [4] R. Conti, Uniformly isochronous centers of polynomial systems in R2, in: Differential Equations, Dynamical Systems, and Control Science, Lecture Notes Pure Appl. Math. 152, Dekker, New York (1994), 21-31.
• [5] N. Debz, A. Boulfoul and A. Berkane, Limit cycles of a class of planar polynomial differential systems, Math. Methods Appl. Sci. 44 (2021), no. 17, 13592-13614.
• [6] N. Debz, A. Boulfoul and A. Berkane, Limit cycles for a class of Kukles type differential systems, Mem. Differ. Equ. Math. Phys. 86 (2022), 31-49.
• [7] M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-Smooth Dynamical Systems: Theory and Applications, Appl. Math. Sci. 163, Springer, London, 2008.
• [8] M. Han and L. Sheng, Bifurcation of limit cycles in piecewise smooth systems via Melnikov function, J. Appl. Anal. Comput. 5 (2015), no. 4, 809-815.
• [9] D. Hilbert, Mathematische Probleme, Gött. Nacht. 1900 (1900), 253-297.
• [10] Z. Jiang, On the limit cycles for continuous and discontinuous cubic differential systems, Discrete Dyn. Nat. Soc. 2016 (2016), Article ID 4939780.
• [11] Y. A. Kuznetsov, S. Rinaldi and A. Gragnani, One-parameter bifurcations in planar Filippov systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 13 (2003), no. 8, 2157-2188.
• [12] C. Li, W. Li, J. Llibre and Z. Zhang, Linear estimation of the number of zeros of abelian integrals for some cubic isochronous centers, J. Differential Equations 180 (2002), no. 2, 307-333.
• [13] S. Li and Y. Zhao, Limit cycles of perturbed cubic isochronous center via the second order averaging method, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 24 (2014), no. 3, Article ID 1450035.
• [14] H. Liang, J. Llibre and J. Torregrosa, Limit cycles coming from some uniform isochronous centers, Adv. Nonlinear Stud. 16 (2016), no. 2, 197-220.
• [15] S. Liu, M. Han and J. Li, Bifurcation methods of periodic orbits for piecewise smooth systems, J. Differential Equations 275 (2021), 204-233.
• [16] J. Llibre and J. Itikawa, Limit cycles for continuous and discontinuous perturbations of uniform isochronous cubic centers, J. Comput. Appl. Math. 277 (2015), 171-191.
• [17] J. Llibre, D. D. Novaes and M. A. Teixeira, On the birth of limit cycles for non-smooth dynamical systems, Bull. Sci. Math. 139 (2015), no. 3, 229-244.
• [18] J. Llibre and G. Świrszcz, On the limit cycles of polynomial vector fields, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 18 (2011), no. 2, 203-214.
• [19] H. Poincaré, Mémoire sur les courbes définies par une équation différentielle. I, J. Math. Pure Appl. 7 (1881), 375-442.
• [20] H. Poincaré, Mémoire sur les courbes définies par une équation différentielle. II, J. Math. Pure Appl. 8 (1882), 251-296.
• [21] H. Poincaré, Sur les courbes définies par les équations différentielle, III, J. Math. Pure Appl. 1 (1885), 167-244.
• [22] H. Poincaré, Sur les courbes définies par les équations différentielle, IV, J. Math. Pure Appl. 2 (1886), 155-217.
• [23] J. Shi, W. Wang and X. Zhang, Limit cycles of polynomial Liénard systems via the averaging method, Nonlinear Anal. Real World Appl. 45 (2019), 650-667.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr POPUL/SP/0154/2024/02 w ramach programu "Społeczna odpowiedzialność nauki II" - moduł: Popularyzacja nauki (2025).